快速求导的策略：幂指数相同时先合并幂指数

一、前言

下面的函数怎么做求导操作，计算速度更快一些：

$$
\begin{aligned}
y_{1} & = \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{3} \\ \\
y_{2} & = \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{6}
\end{aligned}
$$

二、正文

函数 $y_{1}$ 的两种求导方法对比

对于函数 $y_{1}$, 直接进行求导的步骤如下（ 复 杂 ）：

$$
\begin{aligned}
y_{1} ^{\prime} & = \left[ \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{3} \right] ^{\prime} \\ \\
& = 3 \textcolor{tan}{ \left( x-1 \right) }^{2} \cdot \textcolor{tan}{1} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{3} + \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot 3 \textcolor{lightgreen}{ \left( x-2 \right) }^{2} \cdot \textcolor{lightgreen}{1} \\ \\
& = 3 \textcolor{tan}{ \left( x-1 \right) }^{2} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{3} + 3 \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{2} \\ \\
& = 3 \left[ \textcolor{tan}{\left( x-1 \right)}^{2} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{2} \right] \cdot \left[ \textcolor{lightgreen}{\left( x-2 \right)} + \textcolor{tan}{\left( x-1 \right)} \right]
\end{aligned}
$$

对于函数 $y_{1}$, 先合并幂指数，再进行求导的步骤如下（ 简 单 ）：

$$
\begin{aligned}
y_{1} ^{\prime} & = \left[ \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{3} \right] ^{\prime} \\ \\
& = \left\{ \left[ \textcolor{tan}{\left( x-1 \right)} \textcolor{lightgreen}{\left( x-2 \right)} \right]^{3} \right\} ^{\prime} \\ \\
& = 3 \left[ \textcolor{tan}{\left( x-1 \right)} \textcolor{lightgreen}{\left( x-2 \right)} \right]^{2} \cdot \left[ \textcolor{tan}{1} \cdot \textcolor{lightgreen}{\left( x-2 \right)} + \textcolor{tan}{\left( x-1 \right)} \cdot \textcolor{lightgreen}{1} \right] \\ \\
& = 3 \left[ \textcolor{tan}{\left( x-1 \right)} \textcolor{lightgreen}{\left( x-2 \right)} \right]^{2} \cdot \left[ \textcolor{lightgreen}{\left( x-2 \right)} + \textcolor{tan}{\left( x-1 \right)} \right]
\end{aligned}
$$

函数 $y_{2}$ 的两种求导方法对比

对于函数 $y_{2}$, 直接进行求导的步骤如下（ 简 单 ）：

$$
\begin{aligned}
y_{2}^{\prime} & = \left[ \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{6} \right] ^{\prime} \\ \\
& = 3 \textcolor{tan}{\left( x-1 \right)}^{2} \cdot \textcolor{tan}{1} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{6} + \textcolor{tan}{\left( x-1 \right)}^{3} \cdot 6 \textcolor{lightgreen}{\left( x-2 \right)}^{5} \cdot \textcolor{lightgreen}{1} \\ \\
& = 3 \textcolor{tan}{\left( x-1 \right)}^{2} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{6} + \textcolor{tan}{\left( x-1 \right)}^{3} \cdot 6 \textcolor{lightgreen}{\left( x-2 \right)}^{5} \\ \\
& = 3 \left[ \textcolor{tan}{\left( x-1 \right)}^{2} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{5} \right] \cdot \left[ \textcolor{lightgreen}{\left( x-2 \right)} + 2 \textcolor{tan}{\left( x-1 \right)} \right]
\end{aligned}
$$

对于函数 $y_{2}$, 先合并幂指数，再进行求导的步骤如下（ 复 杂 ）：

$$
\begin{aligned}
y_{2}^{\prime} & = \left[ \textcolor{tan}{ \left( x-1 \right) }^{3} \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{6} \right] ^{\prime} \\ \\
& = \left\{ \left[ \textcolor{tan}{ \left( x-1 \right) } \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{2} \right]^{3} \right\} ^{\prime} \\ \\
& = 3 \left[ \textcolor{tan}{ \left( x-1 \right) } \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{2} \right]^{2} \cdot \left[ \textcolor{tan}{1} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{2} + \textcolor{tan}{\left( x-1 \right) } \cdot 2 \textcolor{lightgreen}{\left( x-2 \right)} \cdot \textcolor{lightgreen}{1} \right] \\ \\
& = 3 \left[ \textcolor{tan}{ \left( x-1 \right) } \cdot \textcolor{lightgreen}{ \left( x-2 \right) }^{2} \right]^{2} \cdot \left[ \textcolor{lightgreen}{\left( x-2 \right)}^{2} + \textcolor{tan}{\left( x-1 \right) } \cdot 2 \textcolor{lightgreen}{\left( x-2 \right)} \right] \\ \\
& = 3 \left[ \textcolor{tan}{\left( x-1 \right)}^{2} \cdot \textcolor{lightgreen}{\left( x-2 \right)}^{5} \right] \cdot \left[ \textcolor{lightgreen}{\left( x-2 \right)} + 2 \textcolor{tan}{\left( x-1 \right)} \right]
\end{aligned}
$$

三、总结

总上面的对比可以看出：

对于幂指数相同的乘积因式做求导运算时，先合并（或者说“提取”）幂指数可以简化运算；

对于幂指数不相同的乘积因式做求导运算时，不宜先合并幂指数，而适宜直接进行计算。

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